Integral Methods in Nonlinear Dynamics of Systems

This monograph presents an integral method for analysing the dynamic behaviour of nonlinear, non-stationary, and controlled systems. The method is based on the use of nonlinear integral inequalities to obtain new estimates for the norms of solutions and Lyapunov functions for the corresponding systems of differential equations of disturbed motion. The book consists of seven chapters. The first chapter establishes new bounds for solutions to ordinary and infinite systems of differential equations using nonlinear integral inequalities in pseudo-linear form. The second chapter studies the equations of disturbed motion based on new estimates of Lyapunov functions. Here, conditions are established for various types of motion boundedness, including the stability of coupled systems under initial and subsequent disturbances. The third chapter is devoted to polynomial systems, where variations of Lyapunov functions are used to derive conditions for stability and stabilisation of motion, including the analysis of the zero solution of systems with aftereffects. Chapter 4 applies nonlinear integral inequalities to nonlinear systems with interval initial conditions, and studies the stabilisation of systems with multiple controls. Chapter 5 focuses on quasilinear systems with fractional derivatives, establishing conditions for boundedness and Lagrange stability. Chapter 6 introduces an integral method for time-scale dynamic equations with fractional derivatives, offering new tools for stability and boundedness analysis. The final chapter studies equilibrium stability in a model of confrontation between two countries and alliances, using Lyapunov functions and integral inequalities to determine conditions for stable equilibrium and changes in weapon levels. Sample Chapter(s) Preface References Chapter 7: Equilibrium Stability under Nuclear Confrontation Contents: Preface About the Author Acknowledgements Novel Boundaries for Solutions of Nonlinear Systems with Applications Integral Estimates of Lyapunov Functions and Their Applications Stability and Stabilisation of Motion of Polynomial and Interval Systems Stabilisation of Motions of Interval Bilinear and Affine Systems Stability and Boundedness of Solutions to Fractional-like Equations Elements of Fraction-like Dynamics on the Time Scale Equilibrium Stability under Nuclear Confrontation Appendix A Appendix B References Index